The Jameson Raid was a botched raid against the South African Republic carried out by British colonial statesman Leander Starr Jameson and his Company troops and Bechuanaland policemen over the New Year weekend of 1895–96. Paul Kruger was president of the republic at the time; the raid was intended to trigger an uprising by the British expatriate workers in the Transvaal but failed to do so. The workers were called the Johannesburg conspirators, they were expected to prepare for an insurrection. The raid was ineffective and no uprising took place; the results included embarrassing the British government. What became South Africa was not a single, united nation during the late nineteenth century; the territory had four distinct entities: the two British colonies of Cape Natal. The Cape, more the small area around present day Cape Town, was the first part of South Africa to be settled by Europeans; these settlers were transported by, long remained under the control of, the Dutch East India Company.

Gradual consolidation and eastward expansion took place over the next 150 years. In 1806, Great Britain took over the Cape to prevent the territory falling to Napoleon and to secure control over the trade routes to the Far East. Antipathy towards British control and the introduction of new systems and institutions grew amongst a substantial portion of the Boer community. One of the primary causes of friction was policy of the British authorities towards slavery in South Africa. In 1828, legislation passed by the British parliament guaranteed equal treatment under the law for all, regardless of race. In 1830, a new ordinance imposed heavy penalties for harsh treatment of slaves; the measure was controversial among some of the population, in 1834, the government abolished slavery in the British Empire altogether. The Boers opposed the changes, they believed. They were suspicious of how the government paid for compensation; this resentment culminated in the en-masse migration of substantial numbers of the Boers into the hitherto unexplored frontier, to get beyond the control of British rule.

The migration became known as the Great Trek. This anti-British feeling was by no means universal: in the Western Cape, few Boers felt compelled to move; the Trekboers, frontier farmers in the East, at the front of the colony's eastward expansion, were the ones who elected to trek further afield. These emigrants, or Voortrekkers as they became known, first moved east into the territory known as Natal. In 1839, they founded the Natalia Republic as a new homeland for the Boers. Other Voortrekker parties moved northwards, settling beyond the Vaal rivers. Reluctant to have British subjects moving beyond its control, Britain annexed the Natalia Republic in 1843, which became the Crown colony of Natal. After 1843, British government policy turned against further expansion in South Africa. Although there were some abortive attempts to annex the territories to the north, Britain recognised their independence by the Sand River Convention of 1852 and the Orange River Convention of 1854, for the Transvaal and the Orange Free State, respectively.

After the First Anglo-Boer War, Gladstone's government restored the Transvaal's independence in 1884 by signing the London Convention. No one knew there would be the discovery of the colossal gold deposits of the Witwatersrand two years later. Despite the political divisions, the four territories were linked; each was populated by European-African emigrants from the Cape. As the largest and longest established state in Southern Africa, the Cape was economically and dominant: by comparison, the population of Natal and the two Boer republics were pastoralist, subsistence farmers; the simple agricultural dynamic was upset in 1870, when vast diamond fields were discovered in Griqualand West, around modern-day Kimberley. Although the territory had come under the authority of the Orange Free State, the Cape government, with the assistance of the British government, annexed the area, taking control of its vast mineral wealth. In June 1884, Jan Gerritse Bantjes discovered signs of gold at Vogelstruisfontein followed in September by the Struben brothers at Wilgespruit near Roodepoort which started the Witvatersrand Gold Rush and modern-day Johannesburg.

By 1886 it was clear. The huge inflow of Uitlanders from Britain, had come to the region in search of employment and fortune; the discovery of gold made the Transvaal overnight the richest and the most powerful nation in southern Africa, but it attracted so many Uitlanders that they outnumbered the Boers. Fearful of the Transvaal's losing independence and becoming a British colony, the Boer government adopted policies of protecti

The Mahalanobis distance is a measure of the distance between a point P and a distribution D, introduced by P. C. Mahalanobis in 1936, it is a multi-dimensional generalization of the idea of measuring how many standard deviations away P is from the mean of D. This distance is zero if P is at the mean of D, grows as P moves away from the mean along each principal component axis. If each of these axes is re-scaled to have unit variance the Mahalanobis distance corresponds to standard Euclidean distance in the transformed space; the Mahalanobis distance is thus unitless and scale-invariant, takes into account the correlations of the data set. The Mahalanobis distance of an observation x → = T from a set of observations with mean μ → = T and covariance matrix S is defined as: Mahalanobis distance can be defined as a dissimilarity measure between two random vectors x → and y → of the same distribution with the covariance matrix S: If the covariance matrix is the identity matrix, the Mahalanobis distance reduces to the Euclidean distance.

If the covariance matrix is diagonal the resulting distance measure is called a standardized Euclidean distance: where si is the standard deviation of the xi and yi over the sample set. Mahalanobis distance is preserved under full-rank linear transformations of the space spanned by the data; this means that if the data has a nontrivial nullspace, Mahalanobis distance can be computed after projecting the data down onto any space of the appropriate dimension for the data. We can find useful decompositions of the squared Mahalanobis distance that help to explain some reasons for the outlyingness of multivariate observations and provide a graphical tool for identifying outliers. Consider the problem of estimating the probability that a test point in N-dimensional Euclidean space belongs to a set, where we are given sample points that belong to that set. Our first step would be to find the center of mass of the sample points. Intuitively, the closer the point in question is to this center of mass, the more it is to belong to the set.

However, we need to know if the set is spread out over a large range or a small range, so that we can decide whether a given distance from the center is noteworthy or not. The simplistic approach is to estimate the standard deviation of the distances of the sample points from the center of mass. If the distance between the test point and the center of mass is less than one standard deviation we might conclude that it is probable that the test point belongs to the set; the further away it is, the more that the test point should not be classified as belonging to the set. This intuitive approach can be made quantitative by defining the normalized distance between the test point and the set to be x − μ σ. By plugging this into the normal distribution we can derive the probability of the test point belonging to the set; the drawback of the above approach was that we assumed that the sample points are distributed about the center of mass in a spherical manner. Were the distribution to be decidedly non-spherical, for instance ellipsoidal we would expect the probability of the test point belonging to the set to depend not only on the distance from the center of mass, but on the direction.

In those directions where the ellipsoid has a short axis the test point must be closer, while in those where the axis is long the test point can be further away from the center. Putting this on a mathematical basis, the ellipsoid that best represents the set's probability distribution can be estimated by building the covariance matrix of the samples; the Mahalanobis distance is the distance of the test point from the center of mass divided by the width of the ellipsoid in the direction of the test point. For a normal distribution in any number of dimensions, the probability density of an observation is uniquely determined by the Mahalanobis distance d. D 2 is chi-squared distributed. If the number of dimensions is 2, for example, the probability of a particular calculated d being less than some threshold t is 1 − e − t 2 / 2. To determine a threshold to achieve a particular probability, p, use t = − 2 ln ⁡, for 2 dimensions. For number of dimensions other than 2, the cumulative chi-squared distribution should be consulted.

In a normal distribution, the region where the Mahalanobis distance is less than one is the region where the probability distribution is concav

The Kronstadt rebellion or Kronstadt mutiny was an insurrection of the Soviet sailors and civilians of the port city of Kronstadt against the Bolshevik government of the Russian SFSR. It was the last major revolt against the Bolshevik regime on Russian territory during the Russian Civil War that ravaged the country; the revolt began on March 1, 1921 in the city's naval fortress, located on the island of Kotlin in the Gulf of Finland. Traditionally, Kronstadt served as the base of the Russian Baltic fleet and as defense for the approaches to Petrograd, located 55 kilometres from the island. For sixteen days, the rebels rose in opposition to the Soviet government they had helped to consolidate. Led by Stepan Petrichenko, the rebels, including many Communists disappointed in the direction of the Bolshevik government, demanded a series of reforms, such as the election of new soviets, the inclusion of socialist parties and anarchist groups in the new soviets, the end of the Bolshevik monopoly on power, economic freedom for peasants and workers, dissolution of the bureaucratic organs of government created during the civil war, the restoration of civil rights for the working class.

Despite the influence of some opposition parties, the sailors did not support any in particular. Convinced of the popularity of the reforms they were fighting for, the Kronstadt seamen waited in vain for the support of the population in the rest of the country and rejected aid from emigrants. Although the council of officers advocated a more offensive strategy, the rebels maintained a passive attitude as they waited for the government to take the first step in negotiations, which isolated continental forces; the authorities, by contrast, took an uncompromising stance, presenting an ultimatum demanding unconditional surrender on March 5. Once the surrender period expired, the Bolsheviks sent a series of military raids against the island, managing to suppress the revolt on March 17, causing the deaths of several thousands in the process; the rebels were considered revolutionary martyrs by their supporters and classified as "agents of the Entente and counterrevolution" by the authorities. The Bolshevik response to the revolt caused great controversy and was responsible for the disillusionment of several supporters of the regime established by the Bolsheviks, such as Emma Goldman.

But while the revolt was suppressed and the rebel's political demands were not met, it served to accelerate the implementation of the New Economic Policy, which replaced "war communism". According to Lenin, the crisis was the most critical the regime had yet faced, "undoubtedly more dangerous than Denikin and Kolchak combined". On October 12 the Soviet government signed an armistice with Poland and three weeks the last great White General, Pyotr Nikolayevich Wrangel, abandoned the Crimea, in November the government had managed to disperse Nestor Makhno's Black Army in southern Ukraine. Moscow had regained control of Turkistan and Ukraine, in addition to the coal and oil regions of Donetsk and Baku, respectively. In February 1921, government forces reconquered the Caucasus region with the seizure of Georgia. Although some fighting continued in some regions, these posed no serious military threat to the bolshevik monopoly on power; the government of, having given up hope of a world communist revolution, sought to consolidate power locally and normalize its relations with the Western powers, which ended their intervention in the Russian Civil War.

Throughout 1920 several treaties were signed with other Baltic republics. Despite military victory and improved foreign relations, Russia was facing a serious social and economic crisis, threatening Lenin and his supporters. Foreign troops began to withdraw, yet Bolshevik leaders continued to keep tight control of the economy through the policy of War Communism. Industrial output had fallen dramatically, it is estimated that the total output of mines and factories in 1921 was 20% of the pre-World War I level, with many crucial items suffering an more drastic decline. Production of cotton, for example, had fallen to iron to 2 % of the pre-war level; this crisis coincided with droughts in 1920 and 1921, leading to the Russian famine of 1921. Discontent grew among the Russian populace the peasantry, who felt disadvantaged by Communist grain requisitioning, they resisted by refusing to till their land. In February 1921, more than 100 peasant uprisings took place; the workers in Petrograd were involved in a series of strikes, caused by the reduction of bread rations by one third over a ten-day period.

The revolt at the Kronstadt naval base began as a protest over the plight of the country. By the end of the civil war, Russia was ruined; the conflict had left a large number of victims and the country was plagued by famine and disease. Agricultural and industrial production had been drastically reduced and the transport system was disorganized; the 1920 and 1921 droughts cemented a catastrophic scenario for the country. The arrival of winter and the maintenance of "war communism" and various deprivations by Bolshevik authorities led to increased tensions in the countryside and in the cities Moscow and Petrograd - where strikes and demonstrations took place - in early 1921. Due to the maintenance and reinforcement of "war communism", living conditions worsened eve